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I came across the following problem which says:

Let $a,b,c$ be non-zero real numbers. Also suppose that $\displaystyle f(x)=\sin x,$ if $x \leq c$ and $\displaystyle f(x)=ax+b,$ if $x>c$.Suppose $b$ and $c$ are given. Then which of the following is true ?

$1.$There is no value of $a$ for which $\displaystyle f$ is continuous at $c$
$2.$There is exactly one value of $a$ for which $\displaystyle f$ is continuous at $c$
$3.$There are infinitely many values of $a$ for which $\displaystyle f$ is continuous at $c$
$4.$ Continuity of $\displaystyle f$ at $c$ can not be determined from what is given.

My Attempt: If $\displaystyle f$ is continuous at $c$ ,then we get $a=\frac {\sin c-b}{c}$ using $\lim_{x \to c+}\displaystyle f(x)$ =$\lim_{x \to c-}\displaystyle f(x)$ and this should equal to $\displaystyle f(c).$ Now,I can not progress hereon.

Can someone point me in the right direction? Thanks in advance for your time.

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1 Answer 1

up vote 1 down vote accepted

For continuity, we have to choose $a$ and $b$ so that $ac+b=\sin c$. We have $2$ parameters $a$ and $b$, so there are many choices. There are a lot of straight lines that go through $(c,\sin c)$.

Remark: We should really be discussing all this in terms of limits. But since $\sin x$ and $ax+b$ are continuous everywhere, the limit of $\sin x$ as $x$ approaches $c$ from the left is $\sin c$, and the limit of $ax+b$ as $x$ approaches $c$ from the right is $ac+b$. This gives us the condition for continuity $\sin c=ac+b$ that was used in the answer.

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